057i - Rational Numbers on a Number Line

Lesson 57 — Representing Rational Numbers on a Number Line

Strand: Number | Descriptor: AC9M7N04 | Duration: 45 minutes

Learning Intentions

• To represent rational numbers, including negatives, on a number line.
• To use the number line to compare and order rational numbers.

Success Criteria

I can:

1. Choose a sensible scale for a number line.
2. Place fractions, decimals and percentages accurately on a number line.
3. Place negative rational numbers correctly.
4. Find a rational number lying between any two given numbers.

Warmup

(6 minutes — estimate the position, mini whiteboards)

A number line runs from to , marked only at the ends.

1. Roughly where does sit? How do you know?
2. Where does sit?
3. Where does sit? Is it the same place as ?
4. Where does sit?
5. Which is further right: or ?

Answers: 1. Exactly halfway; 2. A quarter of the way along, halfway between and ; 3. Three quarters along — the same place as , since they are the same number; 4. One tenth along, close to ; 5. , which is slightly further right than .

Activities

Activity 1 — Explicit Instruction: Choosing a Scale (12 min)

The core skill. Before plotting anything, decide what one interval on your line will represent.

The three-step method:

1. Find the range — the smallest and largest numbers you must show.
2. Choose an interval that divides the range sensibly and suits the denominators involved.
3. Label every mark, not just the ends.

I do — plotting , , and .

• Range: all lie between and .
• Denominators involved: , , and decimals to two places. The lowest common denominator of and is , but and need tenths and hundredths.
• Convert everything to decimals first: , , , .
• Use a line from to marked in tenths, and estimate between marks.

The general strategy to state: when a set mixes forms, convert to decimals before plotting. Decimals map directly onto a tenths-and-hundredths scale.

We do — plot on a line from to :

As decimals: , , , , .

Note: and occupy the same point. Students must mark them together, not invent a gap.

You do — plot each set on a suitable line:

1. , , , (line to )
2. , , , (line to )

Activity 2 — Negative Rational Numbers (12 min)

Combines Lesson 11’s integer line with this lesson’s fractions.

The principle. Negative rationals sit to the left of zero, mirroring their positives.

The ordering trap — address it head on. With negatives, the larger the absolute value, the further left the number sits.

This mirrors the integer misconception from Lesson 11 exactly: .

I do. Order from smallest to largest: , , , , .

As decimals: , , , , .

You do — order each set:

1. , , ,
2. , , , ,
3. , , ,

(Answers: 1. ; 2. ; 3. .)

Activity 3 — Inquiry: Numbers in between (10 min)

Pairs.

1. Find a fraction between and .
2. Find a fraction between and your answer to Q1.
3. Keep going. When do you have to stop?
4. How many rational numbers lie between and ?

Socratic scaffolding:

Prompt	Purpose
Understand: what does “between” require?	A number larger than one and smaller than the other.
What is an easy way to find one?	Convert to decimals and pick something in the gap: and give .
Can you do it with fractions directly?	Use a common denominator: and , so sits between.
Another method?	The mean: .
Now repeat between and .	.
And again?	, then — you can always halve the gap.
So when do you stop?	Never. There is always another number in between.
Looking back	Infinitely many rationals lie between any two rationals — no matter how close.

Answers: 1. e.g. ; 2. e.g. ; 3. Never — the process continues indefinitely; 4. Infinitely many.

Contrast to draw out. Between the integers and there are no integers. But between the rationals and there are infinitely many rationals. This property is called density, and it is a genuine difference between the two number systems.

Checks for Understanding

(5 minutes — exit ticket)

1. Place these on a line from to : , , , .
2. Order from smallest to largest: , , , .
3. Find a fraction between and .
4. Reasoning. Explain why is smaller than .
5. How many rational numbers lie between and ?

Answers: 1. , , , in that positional order: ; 2. … converting: , so ; 3. e.g. (since ); 4. lies further left on the number line than , and further left means smaller; 5. Infinitely many.

Common Misconceptions

Misconception	How to pre-empt it
Believing because .	Convert to decimals and locate both on the line. Position decides, not size of digits.
Placing equal numbers written differently at different points.	and mark the same point. Address this in Activity 1.
Spacing marks unevenly on a hand-drawn line.	Use a ruler and equal intervals. An inaccurate line gives wrong conclusions.
Thinking there is a “next” fraction after .	The density inquiry disproves this.
Placing at .	, slightly further right.
Assuming a percentage over cannot be plotted.	is , plotted normally.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Which of these lies closest to on a number line: , , or ?

Answer

Distances from : ; ; . Closest is .

E2 (AMC Junior style). What fraction lies exactly halfway between and ?

Answer

(Check: , , ✓)

E3 (Challenge). On a number line from to , which is further from zero: or ?

Answer

and . So is further from zero — even though it is the smaller number.

E4 (Challenge). Find three different fractions between and .

Answer

Scale up: and . Between them: , , .

E5 (Reasoning challenge). Explain why there is no “smallest positive fraction”.

Answer

Given any positive fraction, halving it gives a smaller positive fraction. Since this can always be done, no smallest one exists. Contrast with the positive integers, where is the smallest.

Homework

1. Draw a number line from to marked in tenths, and plot: , , , , .
2. Draw a number line from to marked in halves, and plot: , , , , .
3. Order from smallest to largest: (a) , , , (b) , , , (c) , , , .
4. Find a fraction between: (a) and (b) and (c) and .
5. Find the number exactly halfway between: (a) and (b) and (c) and .
6. Which is further from zero: (a) or (b) or ?
7. Reasoning. Explain why is greater than , using a number line.
8. Reasoning. Explain why there are infinitely many rational numbers between and .
9. Challenge. Find three fractions between and .

Answers: Q3 — (a) (b) (c) . Q4 — (a) e.g. (b) e.g. (c) e.g. . Q5 — (a) (b) (c) . Q6 — (a) , so it is further (b) both are exactly from zero. Q8 — halving repeatedly always produces a smaller positive number still above zero, so the process never ends. Q9 — scale to eighteenths: and give ; scaling to twenty-sevenths gives , as well.